We know a Posed Closed under suprema but not necessarily under infima is an upper semi-lattice.
We now r.e set forms a distributive lattice.
But my question is why following statement is hold? I think the Turing degree is distributive lattice. I not need a formal proof, just some hint is enough.
Turing Degrees $(D, \leq)$ under the partial order induced by Turing Reducibility $\leq_T$ form only upper semi-lattice.
This is a standard fact given as Corollary 4.4 on page 100 of Soare's book: there exists a pair of degrees with no greatest lower bound. All the details are provided there.